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6.013/ESD.013J Electromagnetics and Applications, Fall 2005
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Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.013 Electromagnetics and Applications
Problem Set #10
Fall Term 2005
Suggested Reading Assignment: Lecture Notes 18
Issued: 11/22/05
Due: 11/30/05
Problem 10.1
A magnetohydrodynamic (MHD) machine of height s and width a is placed within a magnetic
circuit. The fluid moving at velocity V out of the paper obeys Ohm’s law
J = σ (E +V × B)
and has magnetic permeability μo .
(a) A constant dc current i f = I 0 is applied
to the N turn coil. What is the
magnetic field B in the MHD
machine?
(b) What is the voltage v across the load
resistor RL ?
(c) The MHD machine and load resistor RL
are now connected in series with the N
turn coil that has a resistance R f as
μ →∞
if
+
vf
N
s
μo , σ
i
•V
RL
−
shown in the figure to the left.
No current is applied but at t = 0 noise
causes a small current so that
i f ( t = 0 ) = i ( t = 0 ) = I 0 . Neglecting
magnetic saturation of the iron core of
the magnetic current, what is i f ( t )
a
for t ≥ 0 ?
(d) Under what conditions is i f ( t )
Depth D
unbounded as t → ∞ ? Such a system is
called DC self-excited.
Adapted from Problem 6.20 in Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
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Problem 10.2
The field winding of a homopolar generator is connected in series with the rotor terminals
through a capacitor C as shown in the equivalent circuit below. The rotor is turned at constant
speed ω .
(a) For what minimum value of rotor speed is the system self-excited?
(b) For the self-excited condition of (a) what range of values of C will result in dc selfexcitation or in ac self- excitation?
(c) What is the frequency for ac self-excitation?
Problem 6.21 in Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.
Problem 10.3
Figure 6.4.5 in Electromechanical Dynamics, by Herbert H. Woodson and James R. Melcher, 1968.
The electromechanical equivalent circuit of a commutator machine is shown above. The field
and armature windings can be connected in parallel (shunt excitation) or in series (series
excitation) as shown below:
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Figure 6.4.9 in Electromechanical Dynamics, by Herbert H. Woodson and James R. Melcher, 1968.
The electromagnetic torque on the armature (rotor) with moment of inertia J r is T e = Gi f ia .
a) A DC voltage source V0 is connected across the vt terminals for shunt and series
excitations. In the DC steady state, find the terminal current it and torque T e = Gi f ia as a
function of armature angular speed ω = θ& for shunt and series excitations.
b) The voltage source of part (a) that was connected across the vt terminals is now removed
and is replaced by a load resistor RL for shunt and series configurations. Assume that all
circuit variables vary as e st and find natural frequency s for each configuration. For
what rotor speed (ω = θ& ) , direction and magnitude, will the generator be self-excited for
each configuration?
Hint: The shunt excitation algebra can be involved without some forethought. To
simplify the math manipulations for this linear model it is convenient to immediately
write the currents i f and ia as
i f = I f e st
ia = I a e
st
Solve for solutions for s and find unstable roots.
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